Find the 2nd Derivative f(x)=xsin(x)

Problem

ƒ(x)=x*sin(x)

Solution

1. Identify the function ƒ(x)=x*sin(x) and note that finding the second derivative requires differentiating twice with respect to x

2. Apply the product rule to find the first derivative ƒ(x)′ The product rule states (d(u)*v)/d(x)=ud(v)/d(x)+vd(u)/d(x)

3. Differentiate the first time:

ƒ(x)′=xd(sin(x))/d(x)+sin(x)d(x)/d(x)

4. Simplify the first derivative:

ƒ(x)′=x*cos(x)+sin(x)

5. Apply the product rule again to the term x*cos(x) and differentiate sin(x) to find the second derivative ƒ(x)″

6. Differentiate the second time:

ƒ(x)″=(d(x)*cos(x))/d(x)+d(sin(x))/d(x)

ƒ(x)″=(xd(cos(x))/d(x)+cos(x)d(x)/d(x))+cos(x)

7. Simplify the expression:

ƒ(x)″=−x*sin(x)+cos(x)+cos(x)

ƒ(x)″=2*cos(x)−x*sin(x)

Final Answer

ƒ(x)″=2*cos(x)−x*sin(x)

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